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Problem 7.31 PP
where x is the cart position, and the controi input u is a force acting on the cart.
(a) With the state defined as X— [ ^ ^ ^ ^ jT find the feedback gain K that places the 
closed-loop poles at s = -1,-1,-1±iy. For parts (b) through (d), assume that ^ = 0.5.
The normalized equations of motion for an inverted pendulum at angle 0 on a cart are
(b) Use the SRL to select poles with a bandwidth as close as possible to those of part (a), and 
find the control law that will place the closed-loop poles at the points you selected.
(c) Compare the responses of the closed-loop systems in parts (a) and (b) to an initial condition 
of 0 = 10°.You may wish to use the initial command in Matlab.____________________________
(c) Compare the responses of the closed-loop systems in parts (a) and (b) to an initial condition 
of 0 = 10°.You may wish to use the initial command in Matlab.
(d) Compute Nx and Nu for zero steady-state error to a constant command input on the cart 
position, and compare the step responses of each of the two closed-loop systems.
Step-by-step solution
step 1 of 4
A) i ie t[ S I -F + a K )= a .,{ S ) 
Con^aring coefficients yields
_ 10-8P
1 - p 
1 - p
^ = [12 16
A ,=
12]
1 - P
6
Step 2 of 4
B) /f = [0010]
% Symmetric root locus
i= [O ;0 ’G]; 
c = [0*/?.O]; 
d = 0
vlocus (a, b. c, d);
* = [13.5 18.36 3.9 13.98] - 0)
Step 3 of 4 ^
C) The initial condition response to 6 (0)s10°for both control design in (a) and
( i)
Step 4 of 4
D) i W i
Yields
W^=[0 0 1 o f
AT. = 0